Torque proportioning differentials develop a frictional resistance to differentiation between two interconnected drive shafts substantially proportional to a sum of torques transmitted to the drive shafts. During periods of differentiation, the resistance is apparent as a torque difference between the relatively rotating drive shafts. The torque difference varies linearly with the sum of the torques of the two drive shafts defining a substantially constant ratio that can be represented by a symbol "k" as follows: ##EQU1## where "T.sub.d " is the torque difference between the drive shafts, and "T.sub.s " is the torque sum of the two shafts. The symbol k is also a coefficient that represents a proportion of the torque sum T.sub.s that can be developed as a torque difference T.sub.d. Thus, a linear equation for the torque difference T.sub.d can be written as follows: EQU T.sub.d =k T.sub.s.
However, this proportioning characteristic is more commonly expressed as another ratio, referred to as "bias ratio", which is a ratio of respective amounts of torque in the two drive shafts. More particularly, bias ratio represents a quotient of the amount of torque in one of the two relatively rotating drive shafts having more torque divided by the amount of torque in the other relatively rotating drive shaft having less torque. However, bias ratio can also be expressed in terms of the torque difference T.sub.d and torque sum T.sub.s of the drive shafts as follows: ##EQU2## Accordingly, bias ratio "B" is also related to the ratio "k" of the torque difference to torque sum as follows: ##EQU3##
Although most torque proportioning differentials exhibit approximately constant bias ratios over normal ranges of torque transmissions to the drive shafts, some differentials exhibit a bias ratio that decreases with the amount of torque transmitted by the differential. One such example supplements the torque proportioning characteristic with a constant frictional resistance to differentiation, known as "preload", that is independent of the total amount of torque transmitted to the drive shafts. The differential includes a planetary bevel gear arrangement carried within a housing for interconnecting the drive shafts and a pair of friction clutches that are preloaded to provide a constant amount of frictional torque for opposing relative rotation between either drive shaft and the housing.
Torque is also transmitted between the differential housing and drive shafts by a pair of camming members. A shaft that mounts a pair of "spider" gears within the housing includes mating camming surfaces that separate the camming members as a function of the torque transmitted by the differential. The separating movement of the camming members further loads the friction clutches for opposing differential rotation of the drive shafts. Accordingly, the amount of frictional torque developed in opposition to differentiation is also a function of the torque transmitted by the differential.
Accounting for the preload as a constant frictional torque "C.sub.O ", the torque difference T.sub.d that is available to oppose differential rotation between drive shafts is expressed as the following linear equation: EQU T.sub.d =C.sub.0 +k.sub.0 T.sub.s
where coefficient "k.sub.0 " is a proportion of the torque sum T.sub.s that is developed as frictional torque for resisting differentiation.
The bias ratio of this differential example is a function B(T.sub.s) of the torque sum transmitted to the drive shafts and is expressed in equation form as follows: ##EQU4## The calculated bias ratio B(T.sub.s) is most affected by the preload torque C.sub.0 at torque sums T.sub.s that are less than the preload torque. In fact, the bias ratio approaches infinity at very low torque sums. However, the effect of the preload becomes negligible at high amounts of torque transmitted by the differential.
Although it is desirable in certain differential applications to provide an initially high bias ratio that decreases with increasing amounts of torque transmitted by the differential, frictional resistance to differentiation exceeding the amount of transmitted torque (i.e., T.sub.d &gt;T.sub.s) is generally undesirable. The excessive frictional resistance at small amounts of transmitted torque acts as a brake that consumes power, produces unnecessary heat and wear within the differential, and can interfere with anti-lock braking systems.
Another way of varying bias ratio with torque transmitted to a pair of drive shafts is disclosed in U.S. Pat. No. 3,264,900 (HARTUPEE). Torque is transmitted between a differential housing and a conventional bevel gearing arrangement interconnecting the drive shafts by a pair of camming members that are carried within the housing. The camming members have specially contoured surfaces that engage mating spherical surfaces of a spider shaft at angles of friction that increase with increasing amounts of torque transmitted through the camming members. Engagements between the spider shaft and camming members produce separating forces against respective friction clutch assemblies that resist relative rotation between the drive shafts. Although the resistance to differentiation increases with the amount of torque transmitted by the camming members, the resistance decreases as a proportion of the transmitted torque in accordance with the increasing angles of friction on the camming members.
Thus, the proportion of the torque sum T.sub.s that is developed as a torque difference T.sub.d is a function k(T.sub.s) of the torque sum, yielding a torque difference T.sub.d that is expressed most generally by the following nonlinear equation: EQU T.sub.d =k(T.sub.s)T.sub.s
and by appropriate substitution, the bias ratio can also be written in terms of the function k(T.sub.s) as follows: ##EQU5##
Actually, the general function k(T.sub.s) is much broader than that required to support the more limited objective of reducing bias ratio with increasing total torque. Also, the torque proportioning effect of the camming surfaces is very sensitive to amounts of separation between the camming members required to engage the friction clutches. Small amounts of wear in the friction clutches or even ordinary tolerancing variations in the clutches could significantly affect amounts of torque difference supported between the drive shafts at particular amounts of total torque transmitted by the differential.